Branching random walks with alternating sign intensities of branching sources

We consider a continuous-time symmetric branching random walk on a multidimensional lattice with a finite set of particle generation centers, i.e., branching sources. The existence of a positive eigenvalue of the evolutionary operator means the exponential growth of the first moment of the total number of particles both at an arbitrary point and on the entire lattice. Branching random walks with positive or negative intensities of sources that have a simplex configuration are presented in the paper. It is established that the amount of positive eigenvalues of the evolutionary operator, counting their multiplicity, does not exceed the amount of the branching sources with positive intensity, while the maximal eigenvalue is simple. For branching random walk with different positive intensities of sources and arbitrary configuration for both finite and infinite variance of jumps, the critical values of sources' intensities are found, which allows us to prove the existence of positive eigenvalues of the evolutionary operator.